Carlos is using a number line to add another integer to –4. He begins by showing –4 on the number line, as shown below. Carlos adds the other integer on the number line and says that the sum is negative.

Question

A number line going from negative 5 to positive 5. The area between negative 4 to 0 is highlighted.

Which is true of the number Carlos added to –4?
The number must be a negative integer.
The number must be a positive integer.
The number could be 3 or any integer less than 3.
The number could be 4 or any integer greater than 4.

Answer 1

For the sum of –4 and another integer to remain negative, the integer that Carlos added must not be large enough to make the total positive.

Let’s analyze the possible options based on the condition that the sum is negative:

1. The number must be a negative integer.

   • This is true. If Carlos adds a negative integer (e.g., -1, -2, -3), the result will still be negative (e.g., –4 + -1 = –5).

2. The number must be a positive integer.

   • This is false. If Carlos added a positive integer (e.g., +1, +2, +3), it could make the total positive. For example, –4 + 3 = -1, which is negative, but if he added 4, the sum would be 0, and anything greater than 4 would be positive.

3. The number could be 3 or any integer less than 3.

   • This is partially true. While adding 3 could give a negative result (depending on the sign) when added to -4, integers less than 3 (such as -1, 0, or 1) will lead to varying results. However, adding 3 or any integer 5 or greater will make the sum positive.

4. The number could be 4 or any integer greater than 4.

   • This is false. If he added 4 or any integer greater than 4 to -4, the result would definitely be zero or a positive integer.

Considering these analyses, the most accurate statement about the integer Carlos added to –4 is:

The number must be a negative integer.